Process to make textured glass

ABSTRACT

Systems and methods for texturing substrates (e.g., glass, metal, and the like) and the textured substrates produced using such systems and methods are disclosed. An exemplary textured substrate includes a surface having a portion with a root-mean-square roughness between 40 to 1000 microns and an autocorrelation function greater than 0.5 for distances less than 50 microns. An exemplary system for texturing a substrate includes a plunger with a textured surface, where a portion of the textured surface has a root-mean-square roughness between 40 to 1000 microns and an autocorrelation function greater than 0.5 for distances less than 50 microns. An exemplary method for texturing a substrate includes the steps of generating a pattern defining a texture, and 3-D printing the pattern on the substrate to form the texture.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional application of U.S. application Ser.No. 16/269,032, entitled “PROCESS TO MAKE TEXTURED GLASS,” filed on Feb.6, 2019, which claims priority to U.S. Provisional Patent ApplicationNo. 62/627,061, entitled “PROCESS TO MAKE TEXTURED GLASS,” filed on Feb.6, 2018, each of the contents of which are hereby incorporated byreference in their entirety.

FIELD

The present disclosure generally relates to textured glass, and morespecifically to systems and methods for precisely controlling thetexture of textured glass.

BACKGROUND

Traditional methods for texturing glass include pressing molten glasswith textured tools, thereby transferring the texture of the tool ontothe glass. Textures are applied to these tools using traditional methodssuch as sandblasting, electrical discharge machining (EDM), fine/roughmachining, and laser/chemical etching. However, these traditionalmethods are unable to precisely control the texture applied to the tools(e.g., at scales below 500 microns), and thus are unable to preciselycontrol the texture applied to the glass.

In some instances, it is desirable to more precisely control the textureapplied to substrates (e.g., glass). For example, more precise texturecontrol may enable manufacturing of improved glass having low andcontrollable gloss while maintaining high transmissivity (e.g., 80percent transmission). Such glass may be desirable for use in, forexample, rooftop solar panels. In addition, precise control of thetexture of glass can enable improved control of glass appearance, thusallowing the manufacture of glass with desired aesthetic properties(e.g., glass having the appearance of tree bark, roofing material, etc.)while preserving desired optical properties (e.g., low gloss, hightransmissivity, etc.).

BRIEF SUMMARY

Systems and methods for texturing substrates (e.g., glass, metal, andthe like) and the textured substrates produced using such systems andmethods are disclosed. An exemplary textured substrate includes asurface having a portion with a root mean square roughness between 40 to1000 microns and an autocorrelation function greater than 0.5 fordistances less than 50 microns. An exemplary system for texturing asubstrate includes a plunger with a textured surface, where a portion ofthe textured surface has a root-mean-square roughness between 40 to 1000microns and an autocorrelation function greater than 0.5 for distancesless than 50 microns. An exemplary method for texturing a substrateincludes the steps of generating a pattern defining a texture, and 3-Dprinting the pattern on the substrate to form the texture.

FIGURES

FIG. 1 depicts a conventional mold used for texturing substrates.

FIG. 2 depicts a mold used for texturing substrates according to someexamples.

FIG. 3 depicts a mold used for texturing substrates according to someexamples.

FIG. 4 illustrates a process for texturing a substrate according to someexamples.

FIGS. 5A-5D respectively illustrate, an exemplary height mapcorresponding to a texture, a 3-D mesh corresponding to the height map,a 3-D printed metal piece printed using the 3-D mesh, and a glass piecepressed using the metal piece.

FIGS. 6A-6C illustrate exemplary terms that are added to a Hamiltonian,and the respective textures the terms produce when the terms are used inthe Hamiltonian for an exemplary Metropolis-Hastings algorithm.

FIG. 7 illustrates exemplary macroscopic depth modulating functionsf(x,y) and their respective resulting effects on textured substrates.

FIGS. 8A-8D respectively illustrate, an exemplary height map of a “treebark” texture, the height map of the “tree bark” texture that has beenmacroscopically modulated in depth, a 3-D printed metal piece printedusing the macroscopically modulated “tree bark” texture, and a glasspiece pressed using the 3-D printed metal piece.

FIG. 9 illustrates the autocorrelation function for the textures shownin FIGS. 5A-5D.

FIG. 10 illustrates an exemplary autocorrelation function forsandblasted glass.

FIGS. 11A-11B respectively illustrate an exemplary texture generatedusing Fourier transform techniques and the 3-D mesh for the exemplarytexture generated using Fourier transform techniques.

DETAILED DESCRIPTION

The following description is presented to enable a person of ordinaryskill in the art to make and use the various embodiments. Descriptionsof specific systems, devices, methods, and applications are providedonly as examples. Various modifications to the examples described hereinwill be readily apparent to those of ordinary skill in the art, and thegeneral principles defined herein may be applied to other examples andapplications without departing from the spirit and scope of the variousembodiments. Thus, the various embodiments are not intended to belimited to the examples described herein and shown, but are to beaccorded the scope consistent with the claims.

1. Systems for Texturing Substrates

FIG. 1 shows a conventional mold 100 used for texturing substrates(e.g., glass, metal, ceramic, etc.). Mold 100 includes plunger 102including textured surface 104, ring 106, and base 108. To texture asubstrate (e.g., molten glass 110), a press (not shown) applies pressureto plunger 102, causing plunger 102 to press molten glass 110, therebytransferring the texture of textured component 104 (and the texture ofring 106 and of base 108) onto molten glass 110.

As discussed, the texture of textured component 104 is applied usingconventional methods such as sandblasting, electrical dischargemachining (EDM), fine/rough machining, and laser/chemical etching.However, these conventional methods may be unable to precisely controlthe texture applied textured surface 104. For example, sandblastingmethods cannot control the textures applied to substrates at scalesbelow the diameter of the sand grains used and require complex andcumbersome use of masks to obtain spatially varying textures. Chemicaletching methods cannot control the texture at scales below approximately100 microns and are not suitable to obtain spatial variation in texturedepth. Further, sandblasting and chemical etching methods may beundesirable because they can physically weaken substrates.

Further, the optical properties (e.g., the gloss) of substrates (e.g.,glass) produced by these conventional methods may be difficult tocontrol. For example, conventional laser etching methods use laserpulses to etch a surface (e.g., the surface of a roller) for texturingglass. Each laser pulse removes an approximately square or circular area(e.g., of 1600 square microns) with a depth of approximately 50 micronsfrom the surface, resulting in a textured surface, and thus texturedglass, with a step-like or “terrace-like” structure. Because of thisstep-like structure, light impinging on the glass scatters within alimited range of angles, resulting in textured glass with undesirablyhigh gloss. Because conventional laser etching produces glass with suchstep-like structure, conventional laser etching methods are inherentlylimited in their ability to control the gloss of textured glass (e.g.,because of the limited range of scattering angles available on thestep-like surface).

3-D printing using laser sintering to form textures may produce asmoother surface (e.g., a smooth, non-step like surface) on texturedsubstrates. A greater range of scattering angles is available for lightimpinging on substrates with such smoother surface, resulting insubstrates with lower gloss. Additionally, because 3-D printingtechniques are highly controllable (e.g., the texture is defined in aCAD or STL file) on small scales (e.g., scales below 50 microns), 3-Dprinting methods may enable precise user-control of both the opticalproperties (e.g., gloss and transmissivity) and the aesthetic propertiesof textured substrates. Exemplary techniques for generating textures,which are then formed on substrates using 3-D printing are discussedwith respect to FIGS. 4-11 below. Precise control of the texture ofsubstrates produced using the 3-D printing techniques disclosed hereinmay not be achievable by conventional methods of texturing substrates.

FIG. 2 shows an exemplary mold 200 that precisely and controllablytextures substrates (e.g., glass gob 210) according to some examples.Mold 200 includes plunger 202 including textured surface 204, ring 206,and base component 208. In some examples, the texture is formed ontextured surface 204 using 3-D printing techniques (e.g., lasersintering). In some examples, textured surface 204 has a root meansquare roughness between 40 to 1000 microns and an autocorrelationfunction greater than 0.5 for distances less than 50 microns.

In some examples, ring 206 includes textured surface 212 and/or base 208includes textured surface 214. In some examples, the properties (e.g.,the root mean square roughness, the autocorrelation function) oftextured surfaces 212 and 214 are the same as the properties of texturedsurface 204. In other examples, the properties of textured surfaces 212and 214 differ from the properties of textured surface 204. In someexamples, plunger 202, ring 206, base 208, and textured surfaces 204,212, and 214 are made of the same materials such that the components ofmold 200 wear uniformly.

In some examples, the texture of each of the textured surfaces 204, 212,and 214 is defined by a respective pattern (e.g., pattern defined in aCAD or an STL file.) A 3D printing process prints the texture of each ofthe respective patterns onto textured surfaces 204, 212, and 214.

For example, additive manufacturing processes such as laser sintering ofa metal powder onto textured surface 204 forms the texture of surface204 from according to a defined pattern. In some examples, the sinteredmetal powder includes any elemental metal or includes any alloycontaining at least 20 percent in mass of iron, chrome, nickel, cobalt,vanadium, tungsten, molybdenum, aluminum, copper, titanium, platinum,osmium, iridium, or zinc. Exemplary preferred alloys include acobalt-chrome alloy or Inconel. In some examples, textured surfaces 204,212, and 214 are coated with chromium (e.g., a 20 micron layer ofchromium). Coating textured surfaces 204, 212, and 214 with chromium canimprove the respective durability of plunger 202, ring 206 and base 208.

In some examples, any one of textured surfaces 204, 212, and 214 (and/orany one of plunger 202, ring 206, and base 208) are ceramic, and 3Dceramic printing processes form the respective textures of surfaces 204,212, and 214. Having ceramic surfaces 204, 212, and 214 may improve thedurability of mold 200.

In some examples, one or more of plunger 202, ring 206, and base 208,include one or more hollow channels 216. Fluids (e.g., molten salts suchas KCl) circulate through one or more hollow channels 216, enabling heatexchange such that glass gob 210 cools quickly as glass gob 210 is beingpressed by mold 200.

Although textured surfaces 204 and 214 shown in FIG. 2 are generallyflat, it is to be understood that any of textured surfaces 204, 212, and214 can be curved. For example, textured surface 212 is a cylindricalsurface (e.g., the inner surface of ring 206). Further, the processesfor generating and forming the textures of surfaces and substratesdisclosed herein apply regardless of whether the surfaces or substratesare two-dimensional (e.g., a flat plane) or three-dimensional (e.g., aspherical or cylindrical surface).

FIG. 3 depicts exemplary mold 300 for texturing substrates according toother examples. Mold 300 includes three dimensional textured surfaces302 and 304. Mold 300 includes a first roller 306 including texturedsurface 302 and second roller 308 including textured surface 304. Asshown in FIG. 3 , both surfaces 302 and 304 are cylindrical surfaces.The textures of surfaces 302 and 304 are generated and formed usingtechniques analogous to those used to generate and form the textures ofsurfaces 204, 212, and 214 discussed above. In some examples, theproperties of textured surfaces 302 and 304 are identical to theproperties of textured surfaces 204, 212, and 214. For example, texturedsurface 302 has a root mean square roughness between 40 to 1000 micronsand an autocorrelation function greater than 0.5 for distances less than50 microns.

In some examples, any one of first roller 306 and second roller 308include one or more hollow channels 310. Fluids (e.g., molten salts likeKCl) circulate through one or more hollow channels 310, enabling heatexchange so that glass gob 312 cools quickly during pressing by mold300.

2. Techniques for Texturing Substrates

Techniques for texturing substrates (e.g., glass) using molds 200 and300 are now discussed. Referring to FIG. 2 , in some examples, totexture glass gob 210 using mold 200, glass gob 210 is placed upon base208. Pressure is applied from a press (not shown) onto plunger 202,causing plunger 202 to press textured surface 204 into a first surfaceof glass gob 210. The texture of surface 204 is thus applied to thefirst surface of glass gob 210 and the texture of surface 214 is thusapplied to a second opposite surface of glass gob 210. In examples wheremold 200 includes ring 206, the texture of surface 212 is also appliedto a third surface of glass gob 210.

In some examples, the pressure applied by the press onto plunger 202 isbetween 1-100 MPa. In some examples, while being pressed by plunger 202,the temperature of glass gob 210 is between 800° C.-1600° C. In someexamples, the pressure is applied to plunger 202 for a time between0.2-20 seconds.

Referring to FIG. 3 , to texture glass gob 312 using mold 300, in someexamples, a first portion of glass gob 312 is positioned between firstroller 306 and second roller 308. First roller 306 and second roller 308rotate in opposite directions. For example, as shown in FIG. 3 , firstroller 306 rotates counter-clockwise and second roller 308 rotatesclockwise.

After the first portion of glass gob 312 contacts the first roller 306and the second roller 308, glass gob 312 passes between first roller 306and second roller 308, thereby transferring the respective textures ofsurfaces 302 and 304 onto a respective first and second surface of glassgob 312.

In some examples, as glass gob 312 passes between first roller 306 andsecond roller 308, glass gob 312 is at a temperature between 800°C.-1600° C.

3. Techniques for Generating and Forming the Textures of TexturedSubstrates

Techniques for generating and forming the textures of texturedsubstrates (e.g., surfaces 204, 212, 214, 302, and 304 and/or glass gobs210 and 312) are now discussed.

In some examples, the texture of a textured substrate is defined using atwo-dimensional matrix. For example, a height of the texture z(x,y) atcoordinates (x,y) is defined by an index (r,s) of a matrix M. In someexamples, the heights z(x,y) specified in the matrix M are relative to amean height of the matrix. Thus, a matrix M, and a matrix M′=M+C, whereC is a matrix of the same constant, refer to the same texture. A mesh(e.g., a pattern defining the texture) is computed from matrix M (or M′)to produce a file (e.g., a CAD or STL file) that a 3-D printer uses toform the texture on a substrate. In some examples, the 3-D printer formsthe texture on the substrate using the 3-D printing techniques (e.g.,laser sintering of a metal powder) discussed above. Additionally, asdiscussed, in some examples, the textured substrate is coated with aprotective layer (e.g., a 20 micron thick layer of chromium).

As discussed, using 3-D printing to form textures on a substrate may bepreferable over conventional methods of texturing substrates (e.g.,laser etching) because 3D printed textures have a smoother texturedsurface and because 3-D printing is controllable on small scales (e.g.,below 50 microns). A smoother textured surface can enhance the opticalproperties (e.g., lower gloss) of textured substrates. Further, 3-Dprinting can enable the simultaneous printing of both the macroscopicshape of textured substrates and the small-scale textures of texturedsubstrates. This can reduce the need for using multiple steps (e.g.,first machining the substrate and then applying the texture to thesubstrate using conventional methods) when manufacturing texturedsubstrates.

FIG. 4 depicts an exemplary process 400 for texturing a substrate. Atblock 402, a pattern (e.g., a mesh) defining a texture is generated. Insome examples, as discussed below, the pattern has a root mean squareroughness between 40 to 1000 microns and an autocorrelation functiongreater than 0.5 for distances less than 50 microns. At block 404, thepattern is 3-D printed (e.g., using additive laser sintering) onto thesubstrate to form the texture.

Because, in some examples, the pattern is defined by a user-specifiedmatrix, the above discussed systems and techniques enable the ability to“program” individual features (e.g., grains) having a specified shapeand orientation onto textured substrates. Of course, there is no lowerlimit on the resolution with which textures or patterns can be definedusing matrix M. The resolution with which textures can be “programmed”onto substrates using such techniques is thus only limited by theresolution at which 3-D printers can transfer the programmed texturedonto a substrate. Current 3-D printing techniques can transferindividual features of a characteristic length of approximately 25 to 50microns onto substrates. Such programmable and controlled texturing ofsubstrates at the sub 50 micron scale may not be achievable byconventional methods of texturing substrates (e.g., sandblasting, laseretching, chemical etching, machining etc.).

Further, because the disclosed techniques allow a user to “program” atexture onto a substrate, the above techniques enable precise usercontrol over both the optical properties (e.g., reflectivity,transmissivity, gloss level, and the like) and the aesthetic propertiesof textured substrates. For example, using the disclosed techniques, auser can create a mesh defining a pattern (e.g., any set of peaks,valleys, and/or shapes) resembling the appearance of a specified design(e.g., tree bark, roofing tile, a picture) and 3-D print the pattern toform a textured substrate. Additionally, given the characteristics ofthe substrate on which the texture is formed (e.g., the index ofrefraction, reflectance, etc.) and the user-specified pattern, theoptical properties of the resulting textured substrate can be readilydetermined. Accordingly, the disclosed systems and techniques may allowfor a user to program textures onto substrates while balancing desiredoptical performance with desired aesthetic performance.

Exemplary techniques for generating the textures of textured substrates(e.g., generating the matrix M) and the properties of the texturedsubstrates generated using such techniques are now discussed.

An exemplary technique for generating textures uses a stochasticprocesses with a parametric Hamiltonian H. The individual terms of theHamiltonian each correspond to different spatial patterns (e.g.,elliptical patterns, spatially uniform patterns), and by adding terms tothe parametric Hamiltonian, different patterns are generated. Suchstochastic processes can generate textures that appear random, but arecontrolled according to an underlying pattern (e.g., as defined by theparametric Hamiltonian).

An exemplary stochastic process using a parametric Hamiltonian uses aMetropolis-Hastings algorithm to generate patterns. Such processinvolves first initializing a Mo matrix with random values, andcomputing a matrix M_(n+1) from M_(n) by randomly choosing an element(r,s) of matrix M_(n). A matrix A is then generated, where the value ofthe element (r,s) is randomly modified. The parametric Hamiltonians,H(M_(n)) and H(A) are then respectively computed for M_(n) and A. Then,M_(n) or A is chosen the following way:

-   -   with probability        exp(−h(M_(n))/T)/(exp(−h(M_(n))/T)+exp(−h(A)/T)) then        M_(n+1)=M_(n);    -   with probability exp(−h(A)/T)/(exp(−h(M_(n))/T)+exp(−h(A)/T))        then M_(n+1)=A.

Here T is the “temperature” of the system. The “temperature” is chosento be high enough such that the obtained texture appears random, but lowenough such that the obtained texture is not entirely random, butcontrolled according to a pattern. The above process is then repeateduntil a statistically stable configuration is reached. For example, astatistically stable configuration is reached when doubling the numberof iterations in the algorithm does not give a statistically significantdifference in the autocorrelation function of the resultant matrix M(compared to if the number of iterations is not doubled) or does notsignificantly change the global appearance of the texture. In someexamples, 200 iterations are performed for each pixel (e.g., matrixelement). The final matrix M obtained at the last iteration of thealgorithm defines the pattern. In some examples, the matrix M ismultiplied by a coefficient representing a texture depth and the valuesof the matrix are limited to a defined range (e.g., −1 to 1). Forexample, multiplying the generated matrix M (with all values between −1and 1) by a coefficient of 100 microns produces a matrix with anormalized 200 microns depth.

FIG. 5A shows a height map of an exemplary texture generated using theaforementioned Metropolis-Hastings algorithm with the Hamiltonian H1shown in FIG. 6A. Darker regions indicate valleys, while brighterregions indicate peaks. The typical grain size of the texture is 50 to80 microns, and the root mean squared (RMS) depth of the texture is 150microns. FIG. 5B shows the mesh (e.g., pattern) for the texture in FIG.5A, FIG. 5C shows a metal surface of approximately 8 cm×8 cm (e.g.,surface 204) 3-D printed using the mesh in FIG. 5B, and FIG. 5D shows aglass piece (e.g., a textured substrate) of approximately 8 cm×8 cmpressed using the metal surface of FIG. 5C.

As discussed, the individual terms of the Hamiltonian used in theMetropolis-Hastings algorithm each correspond to different spatialpatterns. FIGS. 6A-6C show exemplary Hamiltonians (or exemplary termsthat are included (alone or in combination) in a Hamiltonian), thevalues of parameters used in the corresponding Hamiltonian (and in theMetropolis-Hastings algorithm), the autocorrelation functions of theresulting textures, and the resulting height maps the Hamiltoniansrespectively produce when used in the Metropolis-Hastings algorithm. InFIGS. 6A-6C, (ΔM)_(r,s), is defined as:

$\left( {\Delta M} \right)_{r,s} = \frac{M_{{r + 1},s} + M_{{r - 1},s} + M_{r,{s + 1}} + M_{r,{s - 1}} - {4M_{r,s}}}{\delta^{2}}$where δ is the pixel size. Notably, as shown in FIGS. 6A-6C, theautocorrelation functions for textures generated using H1, H2, and H3are all greater than 0.5 for distances of less than 50 microns.

In some examples, by modulating or changing the values of parameters inthe Hamiltonian, the texture (and/or the orientation of grains formingthe texture) is directionally “stretched.” For example, as shown in FIG.6C, the texture corresponding to the Hamiltonian H3 is “stretched” inthe horizontal direction relative to the texture corresponding to theHamiltonian H2. In this example, changing the value of the parameter “e”in the Hamiltonian H3 causes the stretching. In other examples,depending on the parameters in the Hamiltonian and their respectivevalues, the texture is stretched in the vertical direction (or in anyarbitrary direction). One of ordinary skill in the art will understandthat variable stretching across a texture (both in direction andmagnitude) is obtained by modulating one or more parameters of theHamiltonian (e.g., according to a function) responsible for thestretching effect.

In addition to spatially controlling the texture using the methodsdiscussed above (e.g., by parametrizing the Hamiltonian using the termsin FIGS. 6A-6C), in some examples, the texture is controlled using depthmodulation. In some examples, such depth modulation causes a macroscopicvariation in texture depth (e.g., valleys and peaks) across a texturedsubstrate. Macroscopic variation of texture depth may be desirable tocreate varying levels of gloss across a textured substrate. For example,the below discussed methods of macroscopic depth control produce atextured substrate with a first area having 2 gloss units (measured at60 degrees incidence) and a second area having 8 gloss units (measuredat 60 degrees incidence). Having gloss variation across a texturedsubstrate may be desirable to enhance the aesthetic properties oftextured substrates, especially when viewed from far away. For example,gloss variation across a textured substrate can enable a viewer todistinguish between different portions of the textured substrate,causing the textured substrate to appear “featured” (as opposed touniformly shiny) when viewed from far away.

In some examples, to modulate the depth of a textured substrate, theheight map of the textured substrate z(x.y) (e.g., as defined by matrixM) is multiplied by a function f(x,y) such that z′(x,y) is the heightmap used to generate the pattern for 3D printing andz′(x,y)=f(x,y)z(x,y). In some examples, z′(x,y)=tf(x,y)+f(x,y)z(x,y).f(x,y) is a function representing the macroscopic texture depth andf(x,y) is between 0.1 and 10. t is the depth of a macroscopic patternsuperimposed onto the “microscopic” texture defined by z(x,y). FIG. 7shows exemplary f(x,y) that are used to generate z′(x,y), and theirrespective resultant effect on the macroscopic depth of the texturedsubstrate. For example, the f(x,y) shown in the second row causes thestriped macroscopic variation in texture depth for the texture (e.g.,the height map) shown in the first row and the f(x,y) shown in the thirdrow causes the “tiled” macroscopic variation in texture depth for thetexture shown in the first row.

As discussed, the disclosed depth modulation techniques create texturedsubstrates with areas of different depths (e.g., as measured by rootmean square (RMS)). For example, a first area of the textured substratehas a RMS depth between 20 microns to 1 millimeter, and a second area ofthe textured substrate has an RMS depth differing from the RMS depth ofthe first area by at least 20 percent. In some examples, the first areaand the second area are each at least 4 square millimeters.

FIGS. 8A-8D demonstrate the effect of depth modulation on texturedsubstrates. In particular, FIG. 8A shows a height map of a “tree bark”type texture for an approximately 2 cm×2 cm portion. The “tree bark”type texture is generated by modulating a spatially uniform texture(e.g., shown in FIG. 5A) with a function f(x,y) (e.g., a functioncorresponding to the height map of a picture of a tree bark pattern)representing the macroscopic texture depth variation. FIG. 8B shows theheight map of a spatially uniform texture (e.g., FIG. 5A) modulatedaccording to z′(x,y)=tf(x,y)+f(x,y)z(x,y), where f(x,y) is a functioncorresponding to the height map of a picture of a tree bark pattern. Asshown in FIG. 8B, the macroscopic variation of texture depth (e.g., thedark and light streaks) is superimposed upon the microscopic texturevariation (e.g., the pixels in FIG. 8B). FIG. 8C shows a metal surfaceof approximately 8 cm×8 cm (e.g., surface 204) 3-D printed using thetexture of FIG. 8B. FIG. 8D shows a glass piece (e.g., a texturedsubstrate) of approximately 8 cm×8 cm pressed using the metal surface ofFIG. 8C. As shown in FIG. 8D, different portions of the glass piece havedifferent gloss levels (e.g., some portions appear shinier than others).As shown, this causes the glass piece to appear more featured, as thepeaks and valleys of the glass piece are readily distinguishable.

In some examples, the root mean square (RMS) roughness is used tocharacterize textured substrates (e.g., surfaces 204, 212, 214, 302, and304 or the surfaces of textured glass 210 and 312) and/or the patterndefining the texture. The RMS roughness parameters (e.g., R_(RMS) orR_(q)) are defined according to the BS EN ISO 4287:2000 standard. One ofordinary skill in the art would understand how to measure the RMSroughness (e.g., R_(RMS) or R_(q)) for a textured substrate. In someexamples, a textured substrate (or a portion thereof) produced using thedisclosed techniques has an RMS roughness between 40 to 1000 microns. Insome examples, the preferred RMS roughness is between 40 to 100 microns.

In some examples, the autocorrelation function is used to characterizetextured substrates (e.g., surfaces 204, 212, 214, 302, and 304 or thesurfaces of textured glass 210 and 312) and/or the pattern defining thetexture. To compute the autocorrelation function for a texturedsubstrate, let A be a portion of the textured substrate (e.g., a 25square millimeter portion). An x-y system of Cartesian coordinates isdefined tangent to the average orientation of the textured substrate.The x-y coordinates correspond to length units (e.g., meters). z(x,y) isthe height of the textured substrate (relative to the average height).

Practically, measuring z(x,y) for a textured substrate involvesmeasuring the value of z (e.g., using confocal microscopy) over adiscrete set of values of (x,y). Accordingly, in some examples, thecontinuous function z(x,y) is interpolated from the discretemeasurements of z using bilinear interpolation. In some examples, thecontinuous function z(x,y) is interpolated using a first order finiteelement basis according to a Delaunay triangulation.

To remove the effects of curvature of the substrate and the effects ofthe mean value of z on the autocorrelation function, Z(x,y) is definedas Z(x,y)=z(x,y)−P(x,y). P(x,y)=a+bx+cy+dxy is a first order polynomialfitted using the least squares method to the discrete measured values ofz. Subtracting P(x,y) from z(x,y) thus removes the substrate curvatureeffects and the effects of the mean value of z from Z(x,y). Thus, theaverage of Z(x,y) is 0, and Z(x,y) is on average flat.

For a portion of the textured substrate in the range of x₀<x<x₁ andy₀<y<y₁, the autocorrelation function G(r) for 0<r<min((x₁−x₀)/4),(y₁−y₀)/4), Δx=x₁−x₀, Δy=y₁−y₀ is:

${G(r)} = {\frac{2}{\pi\; r\;\Delta\; x\;\Delta\; y\; R_{RMS}^{2}}{\int_{x_{0} + {\Delta\;{x/4}}}^{x_{0} + {3\;\Delta\;{x/4}}}{{du}\ {\int_{y_{0} + {\Delta\;{y/4}}}^{y_{0} + {3\;\Delta\;{y/4}}}{{dv}{\int_{0}^{2\pi}{d\;\theta\;{Z\ \left( {{\frac{x_{0} + x_{1}}{2} + u},{\frac{y_{0} + y_{1}}{2} + v}} \right)}{Z\left( {{\frac{x_{0} + x_{1}}{2} + u + {r\;{\cos(\theta)}}},{\frac{y_{0} + y_{1}}{2} + v + {r\;{\sin(\theta)}}}} \right)}}}}}}}}$The above autocorrelation function can be calculated using anyreasonable discretized integration method or using Monte-Carlointegration as long as rdθ, du, and dv are each smaller than thehorizontal measurement resolution (i.e., the distance between directlyadjacent (x,y) coordinates for which z is measured). One of skill in theart would understand how to evaluate the above autocorrelation functionfor a textured substrate. For example, the autocorrelation function canbe evaluated using a line scan obtained from a profilometer, the linescan having a length between 1 cm and 5 cm.

In some examples, the techniques discussed above produce texturedsubstrates (and/or patterns defining textured substrates) such that, fora 5 mm×5 mm portion of the textured substrate, G(r)>0.5 for distances rless than 50 microns and G(r)<0.5 for distances r between 2 mm and 2.5mm.

FIG. 9 shows an exemplary autocorrelation function G(r) computed for thetexture shown in FIGS. 5A-5D. As shown, the autocorrelation function inFIG. 9 falls within the above ranges.

The autocorrelation function can be used to distinguish texturedsubstrates produced using the disclosed techniques from texturedsubstrates produced using conventional techniques (e.g., sand blasting,chemical etching). In particular, the autocorrelation functions of flatsubstrates, sandblasted substrates, chemically etched substrates, andperiodically patterned (e.g., with pyramids and grooves) substrates donot fall within the above mentioned ranges. For example, theautocorrelation function of sandblasted glass with a RMS roughness of 4microns is shown in FIG. 10 . As evident upon inspection, theautocorrelation function shown in FIG. 9 (corresponding to a texturedsubstrate formed using the disclosed techniques) and the autocorrelationfunction of FIG. 10 (corresponding to sandblasted glass) are verydifferent. In particular, as shown, G(r) for sandblasted glass is notgreater than 0.5 for r less than 50 microns.

In some examples, Fourier transform methods are used to generate thetexture of textured substrates. Exemplary Fourier transform methodsinclude Gaussian-Free-Field Implementation and phase randomization ofthe Fourier transform of a picture. Generally, exemplary textures areobtained using Fourier transform methods by multiplying the averageamplitude of a Fourier transform by a random value and by a positivereal function corresponding to the desired texture. In particular, toobtain Z(x,y) defining a textured surface, the 2-D Fourier transform ofZ(x,y), F(k_(x),k_(y)), is multiplied by a positive real function ofk_(x) and k_(y) and by a Gaussian random variable for each k_(x) andk_(y). Here, the positive real function of k_(x) and k_(y) correspondsto the underlying patterning of the substrate (as defined in k-space)and the Gaussian random variable causes the features of the finaltextured surface (as defined by Z(x,y)) to appear random. The real partof the inverse Fourier transform of F(k_(x),k_(y)) (multiplied by apositive real function of k_(x) and k_(y) and by a Gaussian randomvariable for each k_(x) and k_(y)) thus defines the final textureZ(x,y).

Substrates generated using the above discussed Fourier transformtechniques are substrates having surfaces with Hurst statistics. Inparticular, F(k) (e.g., F(k_(x),k_(y))) has the form:

-   -   F(k)=0 for k<k₀;    -   F(k)=Ck₁ ^(−2−2H) for k₀<k<k₁;    -   F(k)=Ck^(−2−2H) for k₁<k<k₂; and    -   F(k)=0 for k>k₂.        Here, C is a constant, H is the Hurst exponent, 0<H<3, and        k=√{square root over (k_(x) ²+k_(y) ²)}.

FIG. 11A shows an exemplary height map of a pattern generated using thediscussed Fourier transform techniques. FIG. 11B shows the 3D meshcorresponding to the height map in FIG. 11A. As shown, each “patch”(e.g., a portion of the mesh or height map of approximately the sameheight) has different parameters (e.g., RMS roughness, C value, andHurst exponent). One of skill in the art will understand that theFourier transform F(k_(x),k_(y)) (and thus the C and H values) of atextured substrate can be readily computed from the measured heights ofthe textured substrate.

In some examples, textured substrates generated using the discussedFourier transform techniques have a plurality of “patches” or portionswith different and controllable Hurst exponents. Specifically, thechoice of the positive real function of k_(x) and k_(y) the Fouriertransform F(k_(x),k_(y)) is multiplied by controls the respective Hurstexponent values of the patches.

In some examples, a textured substrate produced using the discussedFourier Transform methods include a plurality (e.g., two or more, threeor more, etc.) of portions, where each portion has a different Hexponent (e.g., differing from the Hurst exponent of every other portionby at least 20 percent). For example, a first portion has a first Hurstexponent, and a second portion has a second Hurst exponent differingfrom the first Hurst exponent by at least 20 percent. In some examples,the first portion and the second portion have respective areas of atleast 4 square millimeters.

In some examples, spatial point processes, such as the Poisson pointprocess, are used to generate the z(x,y) (e.g., the matrix M) definingthe texture for textured substrates. In particular, the Poisson pointprocess chooses randomly, from a set of N random points, a point z_(n)corresponding to x_(n) and y_(n). A localized function f is than addedto that point such that z_(n+1)=z_(n)+f(x−x_(n), y−y_(n)). Similar tothe macroscopic depth modulation techniques discussed above, in someexamples, the localized function f is also modulated to createmacroscopic depth variation across the texture defined by z(x,y).

The foregoing description, for purpose of explanation, has beendescribed with reference to specific examples. However, the illustrativediscussions above are not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Many modifications andvariations are possible in view of the above teachings. The exampleswere chosen and described in order to best explain the principles of theinvention and its practical applications, to thereby enable othersskilled in the art to best utilize the invention and various exampleswith various modifications as are suited to the particular usecontemplated.

Although the disclosure and examples have been fully described withreference to the accompanying drawings, it is to be noted that variouschanges and modifications will become apparent to those skilled in theart. Such changes and modifications are to be understood as beingincluded within the scope of the disclosure and examples as defined bythe claims.

What is claimed is:
 1. A textured substrate, comprising: a first areaand a second area, wherein a Hurst exponent of the first area differsfrom a Hurst exponent of the second area by at least 20 percent, andwherein the first and the second area are at least 4 square millimeters.2. The textured substrate of claim 1, wherein the textured substratecomprises textured glass.
 3. The textured substrate of claim 1, whereinthe textured substrate comprises textured metal.
 4. The texturedsubstrate of claim 3, wherein a texture of the textured substrate isformed by 3-D printing.
 5. The textured substrate of claim 1, whereinthe textured substrate includes a first surface having a first portionthat is textured.
 6. The textured substrate of claim 5, wherein a rootmean square roughness of the first portion is between 40 to 1000microns.
 7. The textured substrate of claim 5, wherein a root meansquare roughness of the first portion is between 40 to 100 microns. 8.The textured substrate of claim 5, wherein the first portion is texturedto include an autocorrelation function greater than 0.5 for distancesless than 50 microns.
 9. The textured substrate of claim 8, wherein theautocorrelation function of the first portion is less than 0.5 fordistances between 2 millimeters and 2.5 millimeters.
 10. The texturedsubstrate of claim 5, wherein a texture of the first portion comprises aplurality of grains, and a user-specified pattern defines the respectiveorientations of the plurality of grains or the respective shapes of theplurality of grains.
 11. The textured substrate of claim 1, wherein thefirst area has a first root mean square depth between 20 microns and 1millimeter, and wherein the second area has a second root mean squaredepth differing from the first root mean square depth by at least 20percent.
 12. The textured substrate of claim 11, wherein the first areahas at least 2 gloss units measured at 60 degrees incidence, and whereinthe second area has at least 8 gloss units measured at 60 degreesincidence.
 13. A method for texturing a substrate to generate thetextured substrate of claim 1, comprising: generating a pattern defininga texture, wherein the pattern has: a root mean square roughness between40 to 1000 microns; and an autocorrelation function greater than 0.5 fordistances less than 50 microns; and 3-D printing the pattern onto thesubstrate to form the texture.
 14. The method of claim 13, wherein 3-Dprinting the pattern onto the substrate includes 3-D printing thepattern onto the substrate using additive laser sintering.